Defining a triangular distribution based on percentiles If I have the 10th percentile, mode and 90% percentile for a triangular distribution, how can I find the minimum (i.e. 0th percentile) and maximum (i.e. 100th percentile) of the distribution?
 A: Using wikipedia's parameterization of the Triangular distribution, it should be apparent that $c$ is the Mode of the distribution. The CDF of this distribution is given by:
$$F(x) = \begin{cases}
0 &, x \leq a \\
\frac{(x-a)^2}{(b-a)(c-a)} &, a < x \leq c \\
1 - \frac{(b-x)^2}{(b-a)(b-c)} &, c < x < b \\
1 & ,x > b
\end{cases}$$
Let's denote the $10^{th}$ and $90^{th}$ percentile of the distribution by $q_{.10}$ and $q_{.90}$ respectively. Then using the CDF we obtain the following system of equations which allows us to solve for $a$ and $b$.
\begin{align*}
0.10 = F(q_{0.10}) \\
0.90 = F(q_{0.90})
\end{align*}

Example
Suppose $Mode = 1$, $q_{0.10} = 0$ and $q_{0.90} = 2.5$. Clearly we have $c = 1$, and to find $a$ and $b$ we write...
\begin{align*}
0.10 &= \frac{(0-a)^2}{(b-a)(1-a)} \\
0.90 &= 1 - \frac{(b-2.5)^2}{(b-a)(b-1)}
\end{align*}
Get out your pen and paper, or just solve for $a$ and $b$ numerically to obtain:
\begin{align*}
a &= -0.93 \\
b &= 3.58 \\
c &= 1
\end{align*}
A: Another way to describe this is to note that the mode occurs when the line changes from a positive to a negative slope. The mode along with the tenth percentile determines the positive slope and the line can be projected down to determine the minimum where the line hits the x-axis.
Similarly, the mode and 90th percentile determine the negative slope which can be used to project the line to the maximum where the line once again hits the x-axis. This is what Big Agnes does algebraically. If the triangle is a right triangle there is only one line. If the slope of the line is negative the minimum will be the mode and the maximum can be determined through the negative slope determined using the mode and the 10th (or 90th) percentile and then projecting down to the x-axis.  Similarly, for a positive slope, the maximum is the mode and the slope is determined by the mode and the 10th (or 90th) percentile with the line projected down to the minimum when it hits the x-axis. These are degenerate cases.
